Numerical solution of equations

Sample questions

Original practice questions written in the style of the exam.

1. Question 16 marks

   $f(x) = x^3 + x - 3$

   (a) Show that $f(x) = 0$ has a root $\alpha$ in the interval $[1, 2]$.

   (b) Use linear interpolation once on the interval $[1, 2]$ to find an approximation to $\alpha$.

   (c) Taking $x_0 = 1.2$, apply the Newton–Raphson procedure once to find a second approximation to $\alpha$, giving your answer to 4 decimal places.

   Mark scheme

   • M1 $f(1) = -1$ and $f(2) = 7$ (1)
   • A1 Sign change and $f$ continuous, so a root lies in $$ (1)